Cornelius (''Niall") O Horgan | University of Virginia (original) (raw)
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Journal Articles by Cornelius (''Niall") O Horgan
Archive for Rational Mechanics and Analysis, 1983
Mechanics of Soft Materials , 2019
We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, ... more We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, namely the occurrence of a "magic angle" associated with the fiber direction which gives rise to special features of the mechanical response. Classically, the magic angle concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. It also arises in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. Such angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible elastic soft solids. In this context, the magic angle concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers. An everyday example of the former is the common garden hose. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle result in quasi-isotropic mechanical response and can lead to material instability.
We identify three distinct shearing modes for simple shear deformations of transversely isotropic... more We identify three distinct shearing modes for simple shear deformations of transversely isotropic soft tissue which allow for both positive and negative Poynting effects (that is, they require compressive and tensile lateral normal stresses, respectively, in order to maintain simple shear). The positive Poynting effect is that usually found for isotropic rubber. Here, specialisation of the general results to three strain-energy functions which are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and consistent with the linear theory suggests that there are two Poynting effects which can accompany the shearing of soft tissue: a dominant negative effect in one mode of shear and a relatively small positive effect in the other two modes. We propose that the relative inextensibility of the fibres relative to the matrix is the primary mechanism behind this large negative Poynting effect.
Papers by Cornelius (''Niall") O Horgan
Journal of Elasticity, 2022
Fiber-reinforcement is a common feature of many soft biological tissues. The response of fibrous ... more Fiber-reinforcement is a common feature of many soft biological tissues. The response of fibrous biotissues to applied shear is thus of considerable current interest. We consider here the fundamental deformation of simple shear within the framework of transversely isotropic incompressible hyperelastic materials for fiber-reinforced specimens with a single family of parallel fibers oriented at a general angle to the direction of shear. It is well known that the normal stress effect characteristic of nonlinear elasticity plays a crucial role in maintaining an homogeneous deformation state in the bulk of the specimen. Here we investigate the effects of fiber-matrix interaction on the shear and lateral normal stresses. The inclusion of fiber-matrix interaction stiffens the shear stress response. As regards the normal stress, it is shown that the confining normal traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy, fiber-matrix interaction and on the angle of orientation of the fibers. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear so that one has the possibility of both a positive or negative Poynting effect. It is shown that the fiber-matrix interaction enhances both the positive and negative Poynting effects. The results are illustrated using experimental data obtained by other authors for porcine brain white matter. In particular, it is shown that, for sufficiently small angles of orientation, a transition amount of shear exists at which the normal stress changes character from tensile to compressive with increasing amount of shear. An increase in the degree of fiber-matrix interaction enlarges the range of orientation angles for which this transition can occur and decreases the transition amount of shear. These transition amounts of shear are well within the physiological strain regime. The results obtained here are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fibrous biological soft tissues.
Journal of Applied Mechanics, 1978
The purpose of this paper is to present new approaches to the problem of wave frequency estimatio... more The purpose of this paper is to present new approaches to the problem of wave frequency estimation for harmonic waves in layered elastic composites. Upper and lower bounds are obtained by adapting standard results for eigenvalue problems with smooth coefficients. The one-dimensional eigenvalue problem with discontinuous coefficients of concern here is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Results are illustrated by numerical examples.
Journal of Applied Mechanics, 1993
The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, fr... more The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, free of tractions on its long sides and subject to edge loads or displacements that produce stresses that decay in the axial direction. If one seeks solutions for the (dimensionless) Airy stress function of the form φ = e−γxF(y), γ constant, then one is led to a fourth-order eigenvalue problem for F(y) with complex eigenvalues γ. This problem, considered previously by Choi and Horgan (1977), is the anisotropic analog of the eigenvalue problem for the Fadle-Papkovich eigenfunctions arising in the isotropic case. The decay rate for Saint-Venant end effects is given by the eigenvalue with smallest positive real part. For an isotropic strip, where the material is described by two elastic constants (Young’s modulus and Poisson’s ratio), the associated eigencondition is independent of these constants. For transversely isotropic (or specially orthotropic) materials, described by four elastic cons...
![Research paper thumbnail of Errata: “Pure azimuthal shear of compressible nonlinearly elastic circular tubes” [Quart. Appl. Math. 52 (1994), no. 1, 113–131; MR1262323 (94m:73052)]](https://attachments.academia-assets.com/89984868/thumbnails/1.jpg)
](Quarterly of Applied Mathematics, 1996
Journal of Applied Mechanics, 1977
Journal of Applied Mechanics, 1982
In this paper, we investigate the predictions of several widely used strain-stiffening phenomenol... more In this paper, we investigate the predictions of several widely used strain-stiffening phenomenological constitutive models for the classical limit point instability that is well- known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons). The shells are composed of incompressible isotropic nonlinearly elastic materials. For a variety of specific strain- energy densities that give rise to strain- stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the monotonicity, or lack thereof, of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two recently developed constitutive models that reflect limiting chain extensibility at the molecular leve...
Series on Advances in Mathematics for Applied Sciences
Archive for Rational Mechanics and Analysis, 1979
Journal of the Mechanical Behavior of Biomedical Materials, 2019
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mechanics Of Soft Materials, 2018
We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, ... more We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, namely the occurrence of a "magic angle" associated with the fiber direction which gives rise to special features of the mechanical response. Classically, the magic angle concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. It also arises in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. Such angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible elastic soft solids. In this context, the magic angle concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers. An everyday example of the former is the common garden hose. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle result in quasi-isotropic mechanical response and can lead to material instability.
Journal of Elasticity, 2018
The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a singl... more The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a single family of oriented parallel fibers is examined. The analysis is purely kinematical and the results are valid for both compressible and incompressible materials. It is shown that for a given amount of shear, for all fiber orientation angles in the range 0 < θ < π/4, the fiber stretch increases with increasing θ whereas in the range π/4 < θ < π/2, this is no longer the case and there is a particular fiber orientation for which the fiber stretch is a maximum. For a particular amount of shear corresponding to a special angle of shear (a "magic" angle of 35.26 •), the fiber-orientation angle at which the fiber stretch is a maximum is its geometric complement namely a magic angle of 54.74 •. The results are also valid for torsion of a circular cylinder reinforced with a single family of helically wound fibers.
IMA Journal of Applied Mathematics, 2014
The phenomenological approach to the modelling of the mechanical response of non-linearly elastic... more The phenomenological approach to the modelling of the mechanical response of non-linearly elastic arteries usually uses a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strainenergy function in favour of a reduced set of only three invariants. It is shown that these anisotropic materials behave as if they were isotropic for two important modes of shearing for cylindrical vessels, indicating possible deficiencies in shear stress predictions.
Journal of Engineering Mathematics, 2021
The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed int... more The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed into an isotropic part representing the mechanical response of an isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the presence of fibres. It is the form of the anisotropic component that is of interest here. We note that the invariants can themselves be divided into two classes: the invariants that are homogeneous functions of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a linear proportional relationship between terms in the general stress–strain law that are of the same degree in the principal stretches. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. The proportionality constants are interpreted as being a measure of the fibre–matrix interaction and a measure of the interaction between fibres in anisotropic non-linear elasticity. An influential model of fibre dispersion is recovered as a special case. The results are illustrated using the homogeneous deformation of simple shear.
Archive for Rational Mechanics and Analysis, 1983
Mechanics of Soft Materials , 2019
We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, ... more We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, namely the occurrence of a "magic angle" associated with the fiber direction which gives rise to special features of the mechanical response. Classically, the magic angle concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. It also arises in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. Such angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible elastic soft solids. In this context, the magic angle concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers. An everyday example of the former is the common garden hose. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle result in quasi-isotropic mechanical response and can lead to material instability.
We identify three distinct shearing modes for simple shear deformations of transversely isotropic... more We identify three distinct shearing modes for simple shear deformations of transversely isotropic soft tissue which allow for both positive and negative Poynting effects (that is, they require compressive and tensile lateral normal stresses, respectively, in order to maintain simple shear). The positive Poynting effect is that usually found for isotropic rubber. Here, specialisation of the general results to three strain-energy functions which are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and consistent with the linear theory suggests that there are two Poynting effects which can accompany the shearing of soft tissue: a dominant negative effect in one mode of shear and a relatively small positive effect in the other two modes. We propose that the relative inextensibility of the fibres relative to the matrix is the primary mechanism behind this large negative Poynting effect.
Journal of Elasticity, 2022
Fiber-reinforcement is a common feature of many soft biological tissues. The response of fibrous ... more Fiber-reinforcement is a common feature of many soft biological tissues. The response of fibrous biotissues to applied shear is thus of considerable current interest. We consider here the fundamental deformation of simple shear within the framework of transversely isotropic incompressible hyperelastic materials for fiber-reinforced specimens with a single family of parallel fibers oriented at a general angle to the direction of shear. It is well known that the normal stress effect characteristic of nonlinear elasticity plays a crucial role in maintaining an homogeneous deformation state in the bulk of the specimen. Here we investigate the effects of fiber-matrix interaction on the shear and lateral normal stresses. The inclusion of fiber-matrix interaction stiffens the shear stress response. As regards the normal stress, it is shown that the confining normal traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy, fiber-matrix interaction and on the angle of orientation of the fibers. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear so that one has the possibility of both a positive or negative Poynting effect. It is shown that the fiber-matrix interaction enhances both the positive and negative Poynting effects. The results are illustrated using experimental data obtained by other authors for porcine brain white matter. In particular, it is shown that, for sufficiently small angles of orientation, a transition amount of shear exists at which the normal stress changes character from tensile to compressive with increasing amount of shear. An increase in the degree of fiber-matrix interaction enlarges the range of orientation angles for which this transition can occur and decreases the transition amount of shear. These transition amounts of shear are well within the physiological strain regime. The results obtained here are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fibrous biological soft tissues.
Journal of Applied Mechanics, 1978
The purpose of this paper is to present new approaches to the problem of wave frequency estimatio... more The purpose of this paper is to present new approaches to the problem of wave frequency estimation for harmonic waves in layered elastic composites. Upper and lower bounds are obtained by adapting standard results for eigenvalue problems with smooth coefficients. The one-dimensional eigenvalue problem with discontinuous coefficients of concern here is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Results are illustrated by numerical examples.
Journal of Applied Mechanics, 1993
The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, fr... more The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, free of tractions on its long sides and subject to edge loads or displacements that produce stresses that decay in the axial direction. If one seeks solutions for the (dimensionless) Airy stress function of the form φ = e−γxF(y), γ constant, then one is led to a fourth-order eigenvalue problem for F(y) with complex eigenvalues γ. This problem, considered previously by Choi and Horgan (1977), is the anisotropic analog of the eigenvalue problem for the Fadle-Papkovich eigenfunctions arising in the isotropic case. The decay rate for Saint-Venant end effects is given by the eigenvalue with smallest positive real part. For an isotropic strip, where the material is described by two elastic constants (Young’s modulus and Poisson’s ratio), the associated eigencondition is independent of these constants. For transversely isotropic (or specially orthotropic) materials, described by four elastic cons...
Quarterly of Applied Mathematics, 1996
Journal of Applied Mechanics, 1977
Journal of Applied Mechanics, 1982
In this paper, we investigate the predictions of several widely used strain-stiffening phenomenol... more In this paper, we investigate the predictions of several widely used strain-stiffening phenomenological constitutive models for the classical limit point instability that is well- known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons). The shells are composed of incompressible isotropic nonlinearly elastic materials. For a variety of specific strain- energy densities that give rise to strain- stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the monotonicity, or lack thereof, of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two recently developed constitutive models that reflect limiting chain extensibility at the molecular leve...
Series on Advances in Mathematics for Applied Sciences
Archive for Rational Mechanics and Analysis, 1979
Journal of the Mechanical Behavior of Biomedical Materials, 2019
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mechanics Of Soft Materials, 2018
We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, ... more We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, namely the occurrence of a "magic angle" associated with the fiber direction which gives rise to special features of the mechanical response. Classically, the magic angle concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. It also arises in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. Such angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible elastic soft solids. In this context, the magic angle concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers. An everyday example of the former is the common garden hose. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle result in quasi-isotropic mechanical response and can lead to material instability.
Journal of Elasticity, 2018
The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a singl... more The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a single family of oriented parallel fibers is examined. The analysis is purely kinematical and the results are valid for both compressible and incompressible materials. It is shown that for a given amount of shear, for all fiber orientation angles in the range 0 < θ < π/4, the fiber stretch increases with increasing θ whereas in the range π/4 < θ < π/2, this is no longer the case and there is a particular fiber orientation for which the fiber stretch is a maximum. For a particular amount of shear corresponding to a special angle of shear (a "magic" angle of 35.26 •), the fiber-orientation angle at which the fiber stretch is a maximum is its geometric complement namely a magic angle of 54.74 •. The results are also valid for torsion of a circular cylinder reinforced with a single family of helically wound fibers.
IMA Journal of Applied Mathematics, 2014
The phenomenological approach to the modelling of the mechanical response of non-linearly elastic... more The phenomenological approach to the modelling of the mechanical response of non-linearly elastic arteries usually uses a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strainenergy function in favour of a reduced set of only three invariants. It is shown that these anisotropic materials behave as if they were isotropic for two important modes of shearing for cylindrical vessels, indicating possible deficiencies in shear stress predictions.
Journal of Engineering Mathematics, 2021
The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed int... more The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed into an isotropic part representing the mechanical response of an isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the presence of fibres. It is the form of the anisotropic component that is of interest here. We note that the invariants can themselves be divided into two classes: the invariants that are homogeneous functions of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a linear proportional relationship between terms in the general stress–strain law that are of the same degree in the principal stretches. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. The proportionality constants are interpreted as being a measure of the fibre–matrix interaction and a measure of the interaction between fibres in anisotropic non-linear elasticity. An influential model of fibre dispersion is recovered as a special case. The results are illustrated using the homogeneous deformation of simple shear.
Journal of Elasticity
The strain-energy density W WWW for incompressible transversely isotropic hyperelastic materials ... more The strain-energy density W WWW for incompressible transversely isotropic hyperelastic materials depends on four independent invariants of the strain tensor. For consistency with the infinitesimal theory, it is well known that there are three necessary conditions on the derivatives of W WWW (evaluated in the undeformed state) that have to be to be satisfied in terms of the three independent elastic moduli of the linear theory for incompressible transversely isotropic materials. We consider three different sets of these linear elastic moduli and express them in terms of the relevant derivatives of W WW W evaluated in the undeformed state. Necessary and sufficient conditions on the linearized strain-energy to ensure positive-definiteness are given and thus we derive such conditions expressed in terms of derivatives of W WWW evaluated in the undeformed state. These conditions are proposed to be fundamental constitutive inequalities which must be satisfied by any physically realistic strain-energy for incompressible transversely isotropic hyperelastic material, in particular those models proposed for fiber-reinforced soft tissues. Some particular strain-energies that have been proposed in the literature are used as illustrations of the results.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
In the analysis of the mechanical behaviour of fibre-reinforced incompressible elastic bodies, th... more In the analysis of the mechanical behaviour of fibre-reinforced incompressible elastic bodies, there is a special angle of orientation of the fibres which leads to a particular mechanical response. This angle has been called a ‘magic angle’ due to its appearance as if by magic in many different aspects of the mechanics of fibrous solids including several examples in biology. It occurs most commonly not only in structural elements composed of circular cylindrical tubes or cylinders reinforced by helically wound fibres but also in flat thin sheets reinforced by fibres in the plane. The occurrence of such a special angle was classically demonstrated using a simple purely geometric analysis in the context of a lattice composed of a single family of helically wound inextensible fibres. Recently, the magic angle concept has been discussed in the framework of nonlinear hyperelasticity for anisotropic materials with detailed constitutive modelling. Our purpose here is to describe some other...
Journal of the Mechanical Behavior of Biomedical Materials
This work is motivated by the current widespread interest in modelling the mechanical response of... more This work is motivated by the current widespread interest in modelling the mechanical response of arterial tissue. A widely used approach within the context of anisotropic nonlinear elasticity is to use an orthotropic incompressible hyperelasticity model which, in general, involves a strain-energy density that depends on seven independent invariants. The complexity of such an approach in its full generality is daunting and so a number of simplifications have been introduced in the literature to facilitate analytical tractability. An extremely popular model of this type is where the strain energy involves only three invariants. While such models and their generalisations have been remarkably successful in capturing the main features of the mechanical response of arterial tissue, it is generally acknowledged that such simplified models must also have some drawbacks. In particular, it is intuitively clear that the correlation of such models with experiment will suffer limitations due to the restricted number of invariants considered. Our purpose here is to use the linearised theory for infinitesimal deformations to provide some guidelines for the development of a more robust nonlinear theory. The linearised theory for incompressible orthotropic materials is developed and involves six independent elastic constants. The general stress-strain law is inverted to provide an expression for the fibre stretch in terms of the stress. We examine the linearised response for simple tension in two mutually perpendicular directions corresponding to the axial and circumferential directions in the artery, obtaining an explicit expression for the fibre stretch in terms of the applied tension, fibre angle and linear elastic constants. The focus is then on determining the range of fibre orientation angles that ensure that the fibres are in tension in these simple tension tests. It is shown that the fibre stretch is positive for both simple tension tests if and only if the fibre angle is restricted to lie between two special angles called generalised magic angles. For the special case where the strain-energy function for the nonlinear model depends only on the three invariants I1,I4,I6, it is shown that the corresponding linearised model, called the standard linear model (SLM), depends on three elastic constants and the fibre stretch is positive only in the small range of fibre angles between the classic magic angles 35.26° and 54.74°. However, when the two additional invariants I5,I7 are included in the nonlinear strain energy so that the corresponding linear model involves four elastic constants, it is shown that the domain of fibre angle for which the stretch is positive is much larger and that the fibre stretch is monotonic with respect to the fibre angle in this range.